Using Genetic Algorithms for Solving the Comparison-Based Identification Problem of Multifactor Estimation Model

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1 Joural of Software Egeerg ad Applcatos, 013, 6, Publshed Ole July 013 ( 349 Usg Geetc Algorthms for Solvg the Comparso-Based Idetfcato Problem of Multfactor Estmato Model Adraws Swda 1, Shmatkov Sergey, Bulav Dmtry 1 Computer Egeerg Departmet, Uversty of Jorda, Amma, Jorda; Computer Scece Departmet, Kharkov Natoal Uversty, Kharkov, Ukrae. Emal: sweda@u.edu.o, tps_kharkov@mal.ru, dmetrod@yadex.ru Receved Aprl 3 rd, 013; revsed May 5 th, 013; accepted Jue d, 013 Copyrght 013 Adraws Swda et al. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. ABSTRACT I ths paper, the statemet ad the methods for solvg the comparso-based structure-parametrc detfcato problem of multfactor estmato model are addressed. A ew method that combes heurstcs methods wth geetc algorthms s proposed to solve the problem. I order to overcome some dsadvatages of usg the classcal utlty fuctos, the use of olear Kolmogorov-Gabor polyomal, whch cotas ts composto the frst as well as hgher characterstcs degrees ad all ther possble combatos s proposed ths paper. The use of olear methods for detfcato of the multfactor estmato model showed that the use of ths ew techque, usg as a utlty fucto the olear Kolmogorov-Gabor polyomal ad the use of geetc algorthms to calculate the weghts, gves a cosderable savg tme ad accuracy performace. Ths method s also smpler ad more evdet for the decso maker (DM) tha other methods. Keywords: Geetc Algorthm; Comparatory Idetfcato; Ftess-Fucto; Chromosome; Crossover; Mutato 1. Itroducto Idetfcato of the obect mathematcal model s to determe ts parameters based o expermetal vestgato of the obect. Idetfcato s the most tme-cosumg ad very mportat operato the sythess model. The classcal problem of detfcato s to determe the mathematcal model y = F(x) of the obect whch cossts of determg the trasformato rules of the put x to output y or more precsely the form ad parameters of operator F. Such detfcato s called drect because t s based o drect quattatve measuremet of put ad output sgals of the obect. However, some cases, there s a eed to detfy a obect, whe the researcher has o drect access to formato about the output sgal. The obects cosdered ths paper, are assumed to be of ths type. I dfferet stuatos, estmates gve by the perso to oe or other propertes of a obect are subectve ad caot be drectly measured by ay physcal devces. I such cases, the classcal methods of the drect detfcato are ot applcable. Alteratve methods are drect detfcato. The most coveet ad wdely used amog these methods s the comparso-based detfcato [1].. Statemet of the Problem Suppose we have a set of alteratves (solutos) X = {x }, 1, m, each of whch s characterzed by a set of dvdual crtera (characterstcs) k, 1,. The values of dvdual crtera k( x) are clearly defed. Based o the aalyss of ths formato a perso shall select the most preferred soluto from the set of solutos X, for example x l,.e. he sets strct order relato o the set of alteratves X: x1 х xm. It meas that, accordg to the utlty theory [], whch postulates the exstece of scalar quatfy the preferece of ay alteratve x X we ca wrte: P x ; P x l l; 1, m, (1) where Px ( ) dvdual scalar evaluato of the usefuless of the alteratves.

2 350 O the bass of ths formato t s ecessary to sythesze the mathematcal model of dvdual choce of the decso maker,.e, a model of geeralzed utlty formato Px ( ). Curretly, the most wdely used two forms of utlty fuctos are: the addtve: ad multplcatve: 1 P x λ k x () P x λ k x (3) where λ somorphsm coeffcets dcatg dmeso, sgfcace, possble values rage, partal crtera k that lead to the somorphsm type. The most formatve stuato s oe whch the coeffcets of somorphsm are gve umercally. Sce λ s a costat, the (3) ca be rewrtte as follows: k λ (4) P x k x Aalyss of (4) shows that the multplcatve estmato does ot take to accout the weghts of partal crtera, sce the product s a costat scalg multplcato factor ad does ot affect the relatoshp of dfferet solutos x X. Therefore, addtve utlty fucto s more uversal ad wdely used. Equato () makes sese oly f takes to accout the mportace of dvdual crtera ad are at the same tme somorphsm coeffcets. Most ofte, defg such coeffcets s a bg problem, so t was decded to represet the addtve utlty fucto the form: P x ak x (5) where a dmesoless relatve weght coeffcets that satsfy the followg restrctos: 0 a 1, a 1, (6) ad k ( x) s ormalzed,.e. trasformed to the somorphc type partal crtera. The ormalzato s performed by the followg formula: k x k x k k k B where k x value of the prvate crtera; k, k B W the best ad worst value (accordgly) of the prvate crtera that s amog the doma of admssble values. I such a way the problem of utlty fucto sythess reduces to the parametrc detfcato of the relatve W W (7) mportace coeffcets. Expert evaluato methods or comparso-based detfcato methods are used for ths purpose [3]. A addtve utlty fucto dsadvatage s that t does ot cosder the possble olear depedece of the utlty fucto o the dvdual crtera absolute values k ad ther mutual fluece. Great theoretcal ad practcal terest s the soluto of the geeral structure-parametrc detfcato problem of the dvdual evaluato model uder less restrctve assumptos about the structure of the model. For ths purpose the Kolmogorov-Gabor polyomal s suggested as a possble structure class. 0 m P x a ak x r g r1 1 g1 ak x * k x 1, ; g 1,, r 1, m, (8) ad geetc algorthms as a method for solvg the geeral structure-parametrc detfcato problem. Ths approach allows us to descrbe ay olear depedece ad does ot mpose ay aprorty restrctos o the addtve or multplcatve utlty fuctos, sce polyomal (8) cotas ts composto the frst as well as hgher degrees of characterstcs k x ad all ther possble combatos. 3. Optmal Complexty Model Defto The am of the soluto the comparatory structure-parametrc detfcato problem s to sythesze a optmal complexty model, whch provdes the mmum error of approxmato crtera of expermetal data output model. Ay sequece of N expermetal data ca be accurately approxmated by a N 1 degree polyomal by solvg a system of ormal algebrac equatos. However, ths approxmato does ot mea that a adequate, hgh accuracy model wth good progostc features s sytheszed. Ths s due to the fact that expermetal data cota measuremet ad other ucotrolled radom errors. Therefore, the polyomal of hgh complexty, ot oly approxmates the desred sgal, but radom errors of expermetal data as well. To overcome ths drawback [3,4] splttg the sample of expermetal data to two sets of data: trag ad testg s proposed. The frst subset s used for the sythess of the model ad determe ts characterstcs, for example the method of least squares, ad the secod to check the accuracy of the model. It was foud that creasg the complexty of the model mproves the accuracy of approxmato of the test sequece of the expermetal data utl t reaches a

3 351 mmum, ad the begs to decle due to the cluso of harmful radom compoets. Model, whch gves a mmum test sequece approxmato error, was amed as the optmal complexty model [3]. Ths rases the problem of choosg crtera of accuracy evaluato of the mathematcal model. I the case of the classcal detfcato the most commoly used crtera s the least squares, for the mplemetato of whch umercal put ad output expermetal data s ecessary. I the case of detfcato of multfactor estmato model, as oted above, quattatve formato about the output effects s ot avalable. I ths regard, a umber of specfc problems, cosdered below, came to the surface. 4. Solvg the Comparatory Idetfcato Problem by the Geetc Algorthms Method I the above formulato, the comparatory structure-parametrc detfcato problem ca be solved by dfferet methods ad algorthms. But commo to all of them s the eed to mplemet a sequece of procedures: geerato of the model structure; defg the quattatve values of ts parameters; assessg the qualty of the model. Varous combatos of algorthms, of dfferet precso, complexty, versatlty, for the frst ad secod stages are possble. To obta perfectess ad versatlty evaluato crtera the feld of the methods applcato there s a eed for ther vestgato. Wth the help of computer expermets geetc programmg algorthms were sytheszed ad vestgated. Geetc Algorthms (GAs) are based o the mechasms of atural selecto ad mplemet a scheme of survval of the fttest amog the cosdered structures, shapg ad chagg the search algorthm based o modelg the evoluto of search. I each geerato a ew set of artfcal sequeces s created usg part of the old set ad the addto of ew parts wth good propertes [5-9]. GA starts wth a radom set of solutos called populato. Each elemet of the populato s called a chromosome ad represets a soluto to the problem. The chromosomes evolve over multple teratos, bearg the ame of geeratos. I the process of terato chromosome s estmated usg the ftess-fucto [6,7,10]. I solvg the problem of structure-parametrc detfcato o the frst step a populato of chromosomes, descrbg the structure of the model s created. Ths s doe by selectg a class of admssble structures. Kolmogorov-Gabor polyomal, takg out the free term ad lmtg t to oly lear ad quadratc terms (squares ad par wse combatos of varables), was chose as ths class. The the polyomal wll be wrtte as follows: C ak xak x ak l x kr x П * 1 l1 l 1, ; r 1,, l r. (9) It meas that for partal crtera the complete polyomal wll have terms, where equal to: C C N C (10) s the umber of combatos ad s!!( )! ( )!! (11) Cosequetly, each chromosome of the populato must cota N bts. The valdty of mposg such lmtatos o the complexty of the polyomal s based o the fact that after the ormalzato by formula (7) all partal characters tcs have values 0 k 1. Squarg these umbers or the multplcato of ay two of them lead to a rapd decrease the values. I addto, each term of the polyomal s multpled by a coeffcet a 1 (because a 1 ). N Based o the fact that the calculato of utlty fucto P X ad weghts a wth accuracy hgher tha two decmals s mpractcal, t ca be cocluded that t s mpractcal as well to clude terms hgher tha the secod order. After the geerato of the chromosomes populato, whch descrbes the model structure, the secod stage, for each of them, a parametrc detfcato s provded by oe of the followg possble methods: Method of determg the Chebyshev pot o the polyhedro descrbed by the system of equaltes P Xl P X, l [5]; The geetc algorthms method. The frst of these methods s descrbed [,5] ad wll ot be cosdered. The mplemetato of the geetc algorthm s as follows. For each chromosome of the populato, whch characterzes a model structure, we determe the umber M of coeffcets a equal to the umber of uts the chromosome. By defto, the coeffcets must satsfy the followg codtos: M 0 a 1, a 1 (1) ad s expressed to two decmal places. Hece the umber of bts that must cota the chromosome of each coeffcet s equal to [8,10]: m 1 b a *10 1, (13) m

4 35 where [a, b ] terval rage of a, poted (1), ad resultat chromosome of all the coeffcets a, 1, M s B L* M (14). For each chromosome of the structure populato we form populato of chromosomes coeffcets a ad solve the problem of the geetc selecto of these coeffcets values that maxmze the match fucto. As a match fucto the umber of satsfed equaltes of (9) ca be take. If ecessary t ca be dvded to trag ad testg sets. O the establshed populatos a teratve procedure of geetc selecto o the frst ad secod populatos to acheve the best value of the match fucto s mplemeted. Example: Let us assume a stuato where a decso maker has to choose the best opto amog fve alteratves of computer systems wth four partal crtera: processor frequecy, memory sze, hard dsk capacty ad prce. The decso maker represets the stuato by costructg Table 1. After that the maxmum ad mmum values of each crtero are defed ad the quattatve ormalzed partal crtera are calculated by formula 7. As a result of the above metoed operatos we get Table whch represets the set of the alteratves wth ther ormalzed partal crtera. Next, the DM selects the best, hs opo, alteratve. Let us assume that the DM chooses the fourth alteratve. Next step s to calculate the addtve utlty fucto. Frst the weght coeffcets are calculated ad the serted the lear Kolmogorov-Gabor polyomal. Thus the addtve utlty fucto s expressed as: Pадд 0.61k 0.39k4 5 (15) We start the procedure of geetc algorthms. Let there be two chromosomes: a paret that cotas the complete Kolmogorov-Gabor polyomal (Fgure 1), ad a chld that cotas oly the compoets of the frst term (Fgure ). Chld chromosome wll be for us the resultat, whch s the shortest polyomal satsfyg the codto R R,? R R R R R R (16) Ths codto (16) wll be the crtero o whch we wll carry out the selecto. Next, usg the above metoed method of geetc algorthms to solve the comparso-based structural-parametrc detfcato we get varat of a chld chromosome that meets crtero (16). The ext step s to choose a optmal legth utlty fucto represeted as the Kolmogorov-Gabor polyomal. That s, the shortest polyomal that satsfes (16) Table 1. The set of alteratves wth ther partal crtera. K 1 K K 3 K 4 R R R R R Table. Alteratves wth ormalzed crtera. K 1 K K 3 K 4 R R R R R Fgure 1. Paret chromosome Fgure. Chld chromosome. ad at the same tme has the maxmum utlty fucto. To do ths, we troduce oe more codto: P P, max (17). ge add Thus, after fdg the optmal legth Kolmogorov- Gabor polyomal satsfyg (16), we check that polyomal. For the chose alteratve substtutg partal crtera we obta the followg legths: P 0.49k 0.571k (18). ge 4 Hece the utlty fuctos of the dfferet alteratves are: R 4 = 1; R 3 = 0.49; R = 0.477; R 5 = 0.04; R1 = The utlty fucto of alteratve R 4 s maxmum, ad other alteratves are worse ad thus the problem s correctly solved. 5. Cocluso The use of olear methods for the detfcato of the multfactor estmato model showed that the use of a ew techque, usg as a utlty fucto the olear Kolmogorov-Gabor polyomal ad the use of the geetc algorthms to calculate the weghts gves a cosderable savg tme ad accuracy performace. It s as well smpler ad more evdet for the decso maker tha other methods. REFERENCES [1] A. O. Ovezgeldyev ad K. E. Petrov, Comparatory Ide-

5 353 tfcato of Lear Multfactor Estmato Models Parameters, Radoelektroka Iformatka, Vol., No. 3, 1998, pp [] A. O. Ovezgeldyev, E. G. Petrov ad K. E. Petrov, Sythess ad Idetfcato of Multfactor Estmato ad Optmzato Models, Naukova Dumka, Kev, 00, 161 p. [3] G. K. Voroovsky, К. V. Machotlo, S. N. Petroshev ad S. А. Sergeev, Geetc Algorthms, Artfcal Neural Networks ad Vrtual Realty Problems, Osova, Kharkov, 1997, 11 p. [4] V. М. Kureychk, Geetc Algorthms. State. Problems, Teorya Systemy Upravleya, No. 1, 1999, pp [5] E. G. Petrov ad Д. А. Булави, Applcato of Chebyshev s Dot ad Geetc Algortms Methods for Determato the Structure of Multfactor Estmato Model, Problemy Bok, No. 58, 003, pp [6] E. G. Petrov ad Д. А. Булави, Applcato of Geetc Algortms Method for Solvg the Multfactor Estmato Model Comparatory Idetfcato Problem, Rado- elektroka Iformatka, No. 1, 003, pp [7] J. Zhag, H. Chug ad W. L. Lo, Clusterg-Based Adaptve Crossover ad Mutato Probabltes for Geetc Algorthms, IEEE Trasactos o Evolutoary Computato, Vol. 11, No. 3, 007, pp do /tevc [8] L. M. Schmtt, Theory of Geetc Algorthms II: Models for Geetc Operators over the Strg-Tesor Represetato of Populatos ad Covergece to Global Optma for Arbtrary Ftess Fucto uder Scalg, Theoretcal Computer Scece, Vol. 310, No. 1, 004, pp do /s (03) [9] D. Goldberg, Geetc Algorthms Search, Optmzato ad Mache Learg, Addso-Wesley Professoal, Readg, [10] А. P. Rotshtey, Itellget Idetfcato Techologes, Uversum-Vca, Vca, 1999, 30 p.